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We study a discrete model of the Laplacian in $\mathbb{R}^2$ that preservesthe geometric structure of the original continual object. This means that,speaking of a discrete model, we do not mean just the direct replacement ofdifferential operators by difference ones but also a discrete analog of theRiemannian structure. We consider this structure on the appropriatecombinatorial analog of differential forms. Self-adjointness and boundness fora discrete Laplacian are proved. We define the Green function for this operatorand also derive an explicit formula of the one.

Green function for a two-dimensional discrete Laplace-Beltrami operator